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Let A = QXQ and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for all (a, b), (c, d) belongs to A then find identity element in A.

Let A = QXQ and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for all (a, b), (c, d) belongs to A then find identity element in A. Correct Answer (1, 0)

Concept:

Let A = QXQ and let * be a binary operation on A and e = (a', b') be the identity element of A.

⇒ p * e = p =  e * p

 

Calculations:

Let A = QXQ and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for all (a, b), (c, d) belongs to A.

Let e = (a', b') be the identity element of A.

⇒ p * e = p =  e * p

Consider, p * e = p

(a, b) * (a', b') = (a, b)

⇒ (aa', b + ab') = (a, b)

⇒ aa' = a, b + ab' = b

⇒ a' = 1, b' = 0

⇒ e = (1, 0) be the identity element of A.

Hence, A = QXQ and let * be a binary operation on A defined by (a, b) * (c, d) = (ac, b + ad) for all (a, b), (c, d) belongs to A then the identity element in A is (1, 0)

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